Sorry missed your reply. The mathematical formulation is as follow:

Let

be the counts of each ball picked by person 1, where is the number of picks and is the number of balls in the bag. In particular, and in your question.

Similarly we can let be another multinomial vector representing the picks from person 2, which has the identical distribution as and they are independent.

The number of matches, by definition, is

and you want to find out the distribution of this discrete random variable.

I have not find a nice method to tackle this problem yet; just use R to simulate 1 million times to obtain a numerical solution first.

where the last line gives you the estimate of probability mass function from left to right.